To extract all the available positional information requires mechanisms that preserve eight or more bits of information about the combined expression levels of the four gap genes. We can visualize what is being gained using the decoding maps introduced in Ref [1]}. At the top of fig REF C we consider cells at position \(x\)  that have access to the expression levels of all four gap genes, and use these data to infer their position \(x^{\prime }\) ; gray levels show the probability distributions \(P(x^{\prime }|x)\) , which in this case form a narrow band around \(x^{\prime }=x\) , with width \(\sigma _x/L \sim 0.01\) . In the lower panels we imagine that inference is based not on the actual expression levels but on the compressed version \(C\)  that emerges from solving the bottleneck problem, and we do this for the optimal compressions with \(I(C;\lbrace g_{\rm i}\rbrace ) = 2,\,\)  and \(4\,{\rm bits}\) . We see that as the compression becomes more severe, the inference becomes more uncertain (larger \(\sigma _x\) ) and genuinely ambiguous.
